Now consider the composed operator \(\Pi _{\kappa _b}\mathcal {T}_\pi ^n(\cdot )\) . Observe that the \(n\) -step TD operator \(\mathcal {T}_\pi ^n(\cdot )\)  is a contraction mapping with respect to \(\Vert \cdot \Vert _\infty \) , with contraction factor \(\gamma ^n\) . Since the contraction factor of \(\mathcal {T}_\pi ^n(\cdot )\)  decreases geometrically fast as \(n\)  increases, by choosing \(n\)  large enough, one can ensure that \(\Pi _{\kappa _b}\mathcal {T}_\pi ^n(\cdot )\)  is a contraction with respect to any chosen norm. This important observation enables us to establish the convergence of Algorithm REF  in Section REF . A similar idea was exploited in off-policy TD\((\lambda )\)  algorithm in [1]}, where it was shown that for \(\lambda \)  close to unity, the off-policy TD\((\lambda )\)  algorithm converges. However, [1]} requires an additional projection step in the algorithm to establish the convergence, and no finite-sample guarantees were shown.
